PART 2Fuzzy sets vs crisp sets

1. Properties of α-cuts2. Fuzzy set representations3. Extension principle

FUZZY SETS AND

FUZZY LOGICTheory and Applications

Properties of α-cuts

• Theorem 2.1

Let A, B F(X). Then, the following properties hold for all α, β [0, 1]:

(i)

(ii)

(iii)

(iv)

(v)

)(

;)(and )(

;)(and )(

;andimplies

;

)1( AA

BABABABA

BABABABA

AAAA

AA

Properties of α-cuts

Properties of α-cuts

• Theorem 2.2

Let Ai F(X) for all i I, where I is an index set. Then,

(vi)

(vii)

.)( and )(

;)( and )(

Ii Iiii

Iii

Iii

Ii Iiii

Iii

Iii

AAAA

AAAA

Properties of α-cuts

•

)(

)(

1)(sup))((But

arbitrary ,1 , ,1

1)(Let

11

1

1

1

Iii

Iii

Iii

Iii

iIiIi

i

Iii

i

i

AA

XA

XA

xAxA

A

A

XNIi

xA

Properties of α-cuts

• Theorem 2.3

Let A, B F(X). Then, for all α [0, 1],

(viii)

(ix)

.iff

;iff

;iff

;iff

BABA

BABA

BABA

BABA

Properties of α-cuts

• Theorem 2.4

For any A F(X), the following properties hold:

(x)

(xi)

;.

;

AAA

AAA

Fuzzy set representations

• Theorem 2.5 (First Decomposition Theorem)

For every A F(X),

where αA is defined by (2.1), and ∪ denotes the standard fuzzy union.

]1,0[

,

AA

Fuzzy set representations

Fuzzy set representations

• Theorem 2.6 (Second Decomposition Theorem)

For every A F(X),

where α+A denotes a special fuzzy set defined by

and ∪ denotes the standard fuzzy union.

]1,0[

,

AA

)()( xAxA

Fuzzy set representations

• Theorem 2.7 (Third Decomposition Theorem)

For every A F(X),

where Λ(A) is the level of A, αA is defined by (2.1), and ∪denotes the standard fuzzy union.

)(

,A

AA

Fuzzy set representations

Extension principle

Extension principle

• Extension principle.

Any given function f : X→Y induces two functions,

),()( :

),()( :1 XYf

YXf

FF

FF

Extension principle

which are defined by

for all A F(X) and

for all B F(Y).

)(sup))](([)( |xAyAf

xfyx

))(())](([ 1 xfBxBf

Extension principle

Extension principle

Extension principle

• Theorem 2.8

Let f : X→Y be an arbitrary crisp function. Then, for any Ai F(X) and any Bi F(Y),

i I, the following properties of functions obtained by the extension principle hold:

Extension principle

Extension principle

• Theorem 2.9

Let f : X→Y be an arbitrary crisp function.

Then, for any Ai F(X) and all α [0, 1]

the following properties of fuzzified by the

extension principle hold:

Extension principle

•

xxA

xb

xaxf

NX

11)(

10 ,

10 , )(

,1 ,Let

Extension principle

Extension principle

•

)].([ )(

00)(But

10)]([

1)(sup))](([

9.0)(sup))](([

11

1

1

|

|

AfAf

bafA

baAf

xAbAf

xAaAf

bxx

axx

Extension principle

• Theorem 2.10

Let f : X→Y be an arbitrary crisp function.

Then, for any Ai F(X), f fuzzified by the

extension principle satisfies the equation:

]1,0[

)()(

AfAf

Exercise 2

• 2.4

• 2.8

• 2.11